Abstract

A modeling procedure is demonstrated, which allows representation of polarization-resolved BRDF data using only four parameters: the real and imaginary parts of an effective refractive index with an added parameter taking grazing incidence absorption into account and an angular-scattering parameter determined from the BRDF measurement of a chosen angle of incidence, preferably close to normal incidence. These parameters allow accurate predictions of s- and p-polarized BRDF for a painted rough surface, over three decades of variation in BRDF magnitude. To characterize any particular surface of interest, the measurements required to determine these four parameters are the directional hemispherical reflectance (DHR) for s- and p-polarized input radiation and the BRDF at a selected angle of incidence. The DHR data describes the angular and polarization dependence, as well as providing the overall normalization constraint. The resulting model conserves energy and fulfills the reciprocity criteria.

© 2011 OSA

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References

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  1. K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57(9), 1105–1114 (1967).
    [CrossRef]
  2. J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional reflectance model validation and utilization,” Technical Report AFAL-TR-73–303, Environmental Research Institute of Michigan (ERIM), October 1973.
  3. J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE 3699, 2–15 (1999).
    [CrossRef]
  4. P. Beckmann and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces,” (Pergamon Press, 1963).
  5. X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Comput. Graph. 25(4), 175–186 (1991).
    [CrossRef]
  6. D. B. Goldman, B. Curless, A. Hertzmann, and S. M. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 32(6), 1060–1071 (2010).
    [CrossRef] [PubMed]
  7. R. Priest, and T. Germer, “Polarimetric BRDF in the microfacet model: theory and measurements,” Proc. Military Sensing Symposia Specialty Group on Passive Sensors, Vol. 1, pp. 169–181 (Infrared Information Analysis Center, Ann Arbor, MI, August 2000), available at http://physics.nist.gov/Divisions/Div844/publications/germer/ GermerPriestMicroFacet.pdf
  8. T. A. Germer and E. Marx, “Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings,” Appl. Opt. 43(6), 1266–1274 (2004).
    [CrossRef] [PubMed]
  9. I. G. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express 16(17), 12892–12898 (2008).
    [CrossRef] [PubMed]
  10. J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s Law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998).
    [CrossRef]
  11. J. Stover, Optical Scattering, Measurement and Analysis (SPIE Press, 1995).
  12. B. G. Hoover and V. L. Gamiz, “Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics,” J. Opt. Soc. Am. A 23(2), 314–328 (2006).
    [CrossRef]
  13. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley-Interscience, New York, 1983).
  14. R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
    [CrossRef]
  15. F. Nicodemus, J. Richmond, J. Hsia, I. Ginsburg, and T. Lamparis, “Geometrical considerations and nomenclature for reflectance,” Nat. Bur. Stand. (U.S.) Monograph 160 (1977).
  16. D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998).
    [CrossRef]
  17. M. J. Persky and M. Szczesniak, “Infrared, spectral, directional-hemispherical reflectance of fused silica, Teflon polytetrafluoroethylene polymer, chrome oxide ceramic particle surface, Pyromark 2500 paint, Krylon 1602 paint, and Duraflect coating,” Appl. Opt. 47(10), 1389–1396 (2008).
    [CrossRef] [PubMed]
  18. H. Holl, “Specular reflection and characteristics of reflected light,” J. Opt. Soc. Am. 57(5), 683–690 (1967).
    [CrossRef]

2010

D. B. Goldman, B. Curless, A. Hertzmann, and S. M. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 32(6), 1060–1071 (2010).
[CrossRef] [PubMed]

2008

I. G. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express 16(17), 12892–12898 (2008).
[CrossRef] [PubMed]

M. J. Persky and M. Szczesniak, “Infrared, spectral, directional-hemispherical reflectance of fused silica, Teflon polytetrafluoroethylene polymer, chrome oxide ceramic particle surface, Pyromark 2500 paint, Krylon 1602 paint, and Duraflect coating,” Appl. Opt. 47(10), 1389–1396 (2008).
[CrossRef] [PubMed]

2006

B. G. Hoover and V. L. Gamiz, “Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics,” J. Opt. Soc. Am. A 23(2), 314–328 (2006).
[CrossRef]

2004

T. A. Germer and E. Marx, “Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings,” Appl. Opt. 43(6), 1266–1274 (2004).
[CrossRef] [PubMed]

2002

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[CrossRef]

1999

J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE 3699, 2–15 (1999).
[CrossRef]

1998

D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998).
[CrossRef]

J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s Law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998).
[CrossRef]

1991

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Comput. Graph. 25(4), 175–186 (1991).
[CrossRef]

1967

H. Holl, “Specular reflection and characteristics of reflected light,” J. Opt. Soc. Am. 57(5), 683–690 (1967).
[CrossRef]

K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57(9), 1105–1114 (1967).
[CrossRef]

Blasband, C.

J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE 3699, 2–15 (1999).
[CrossRef]

Boreman, G. D.

I. G. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express 16(17), 12892–12898 (2008).
[CrossRef] [PubMed]

Bruegge, C. J.

D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998).
[CrossRef]

Curless, B.

D. B. Goldman, B. Curless, A. Hertzmann, and S. M. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 32(6), 1060–1071 (2010).
[CrossRef] [PubMed]

Duval, V.

D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998).
[CrossRef]

Gamiz, V. L.

B. G. Hoover and V. L. Gamiz, “Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics,” J. Opt. Soc. Am. A 23(2), 314–328 (2006).
[CrossRef]

Germer, T. A.

T. A. Germer and E. Marx, “Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings,” Appl. Opt. 43(6), 1266–1274 (2004).
[CrossRef] [PubMed]

Goldman, D. B.

D. B. Goldman, B. Curless, A. Hertzmann, and S. M. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 32(6), 1060–1071 (2010).
[CrossRef] [PubMed]

Greenberg, D. P.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Comput. Graph. 25(4), 175–186 (1991).
[CrossRef]

Greffet, J.

J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s Law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998).
[CrossRef]

Haner, D. A.

D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998).
[CrossRef]

He, X. D.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Comput. Graph. 25(4), 175–186 (1991).
[CrossRef]

Hertzmann, A.

D. B. Goldman, B. Curless, A. Hertzmann, and S. M. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 32(6), 1060–1071 (2010).
[CrossRef] [PubMed]

Hilgers, J. W.

J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE 3699, 2–15 (1999).
[CrossRef]

Holl, H.

H. Holl, “Specular reflection and characteristics of reflected light,” J. Opt. Soc. Am. 57(5), 683–690 (1967).
[CrossRef]

Hoover, B. G.

B. G. Hoover and V. L. Gamiz, “Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics,” J. Opt. Soc. Am. A 23(2), 314–328 (2006).
[CrossRef]

Jafolla, J. C.

J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE 3699, 2–15 (1999).
[CrossRef]

Marx, E.

T. A. Germer and E. Marx, “Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings,” Appl. Opt. 43(6), 1266–1274 (2004).
[CrossRef] [PubMed]

McGuckin, B. T.

D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998).
[CrossRef]

Meier, S. R.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[CrossRef]

Menzies, R. T.

D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998).
[CrossRef]

Nieto-Vesperinas, M.

J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s Law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998).
[CrossRef]

Persky, M. J.

M. J. Persky and M. Szczesniak, “Infrared, spectral, directional-hemispherical reflectance of fused silica, Teflon polytetrafluoroethylene polymer, chrome oxide ceramic particle surface, Pyromark 2500 paint, Krylon 1602 paint, and Duraflect coating,” Appl. Opt. 47(10), 1389–1396 (2008).
[CrossRef] [PubMed]

Priest, R. G.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[CrossRef]

Renhorn, I. G.

I. G. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express 16(17), 12892–12898 (2008).
[CrossRef] [PubMed]

Reynolds, W. R.

J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE 3699, 2–15 (1999).
[CrossRef]

Seitz, S. M.

D. B. Goldman, B. Curless, A. Hertzmann, and S. M. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 32(6), 1060–1071 (2010).
[CrossRef] [PubMed]

Sillion, F. X.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Comput. Graph. 25(4), 175–186 (1991).
[CrossRef]

Sparrow, E. M.

K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57(9), 1105–1114 (1967).
[CrossRef]

Szczesniak, M.

M. J. Persky and M. Szczesniak, “Infrared, spectral, directional-hemispherical reflectance of fused silica, Teflon polytetrafluoroethylene polymer, chrome oxide ceramic particle surface, Pyromark 2500 paint, Krylon 1602 paint, and Duraflect coating,” Appl. Opt. 47(10), 1389–1396 (2008).
[CrossRef] [PubMed]

Thomas, D. J.

J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE 3699, 2–15 (1999).
[CrossRef]

Torrance, K. E.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Comput. Graph. 25(4), 175–186 (1991).
[CrossRef]

K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57(9), 1105–1114 (1967).
[CrossRef]

Appl. Opt.

T. A. Germer and E. Marx, “Ray model of light scattering by flake pigments or rough surfaces with smooth transparent coatings,” Appl. Opt. 43(6), 1266–1274 (2004).
[CrossRef] [PubMed]

D. A. Haner, B. T. McGuckin, R. T. Menzies, C. J. Bruegge, and V. Duval, “Directional-hemispherical reflectance for spectralon by integration of its bidirectional reflectance,” Appl. Opt. 37(18), 3996–3999 (1998).
[CrossRef]

M. J. Persky and M. Szczesniak, “Infrared, spectral, directional-hemispherical reflectance of fused silica, Teflon polytetrafluoroethylene polymer, chrome oxide ceramic particle surface, Pyromark 2500 paint, Krylon 1602 paint, and Duraflect coating,” Appl. Opt. 47(10), 1389–1396 (2008).
[CrossRef] [PubMed]

Comput. Graph.

X. D. He, K. E. Torrance, F. X. Sillion, and D. P. Greenberg, “A comprehensive physical model for light reflection,” Comput. Graph. 25(4), 175–186 (1991).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

D. B. Goldman, B. Curless, A. Hertzmann, and S. M. Seitz, “Shape and spatially-varying BRDFs from photometric stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 32(6), 1060–1071 (2010).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

K. E. Torrance and E. M. Sparrow, “Theory for off-specular reflection from roughened surfaces,” J. Opt. Soc. Am. 57(9), 1105–1114 (1967).
[CrossRef]

H. Holl, “Specular reflection and characteristics of reflected light,” J. Opt. Soc. Am. 57(5), 683–690 (1967).
[CrossRef]

J. Opt. Soc. Am. A

J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s Law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998).
[CrossRef]

B. G. Hoover and V. L. Gamiz, “Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics,” J. Opt. Soc. Am. A 23(2), 314–328 (2006).
[CrossRef]

Opt. Eng.

R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002).
[CrossRef]

Opt. Express

I. G. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express 16(17), 12892–12898 (2008).
[CrossRef] [PubMed]

Proc. SPIE

J. C. Jafolla, D. J. Thomas, J. W. Hilgers, W. R. Reynolds, and C. Blasband, “Theory and measurement of bidirectional reflectance for signature analysis,” Proc. SPIE 3699, 2–15 (1999).
[CrossRef]

Other

P. Beckmann and A. Spizzichino, “The Scattering of Electromagnetic Waves from Rough Surfaces,” (Pergamon Press, 1963).

J. R. Maxwell, J. Beard, S. Weiner, D. Ladd, and S. Ladd, “Bidirectional reflectance model validation and utilization,” Technical Report AFAL-TR-73–303, Environmental Research Institute of Michigan (ERIM), October 1973.

R. Priest, and T. Germer, “Polarimetric BRDF in the microfacet model: theory and measurements,” Proc. Military Sensing Symposia Specialty Group on Passive Sensors, Vol. 1, pp. 169–181 (Infrared Information Analysis Center, Ann Arbor, MI, August 2000), available at http://physics.nist.gov/Divisions/Div844/publications/germer/ GermerPriestMicroFacet.pdf

F. Nicodemus, J. Richmond, J. Hsia, I. Ginsburg, and T. Lamparis, “Geometrical considerations and nomenclature for reflectance,” Nat. Bur. Stand. (U.S.) Monograph 160 (1977).

C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley-Interscience, New York, 1983).

J. Stover, Optical Scattering, Measurement and Analysis (SPIE Press, 1995).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the pBRDF model. The red arrow shows the direction to the source and the blue arrow illustrates a scattering direction.

Fig. 2
Fig. 2

Hemispherical (DHR) data for s- and p-polarizations at 3.39 µm as a function of angle of incidence for the green paint. The fitted values were σ N 0 = 0.4528, n = 1.526 and k = 0.193.

Fig. 3
Fig. 3

Variations of the roughness parameter ρq for s- (blue curve) and p-polarizations (red curve).

Fig. 4
Fig. 4

The pBRDF model was fitted to measurements using non-linear least-squares method resulting in ρ0 = 0.473 at θi = 0 degrees(a) and ρ0 = 0.457 at θi = 20 degrees (b).

Fig. 5
Fig. 5

Scattering from a painted surface illuminated by s-polarized radiation at 3.39 µm together with predictions based on the four parameter solution. The predictions are shown for θi = 0, 20 and 40 degrees (a) and θi = 60 and 80 degrees (b).

Fig. 6
Fig. 6

Scattering from a painted surface illuminated by p-polarized radiation at 3.39 µm together with predictions based on the four parameter solution. The predictions are shown for θi = 0, 20 and 40 degrees (a) and θi = 60 and 80 degrees (b).

Equations (15)

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f B R D F ( θ i , φ i , θ r , φ r , λ ) = L ( θ r , φ r ) E ( θ i , φ i ) [ s r 1 ]
D H R q ( α i , λ ) = 1 α 2 1 α 2 1 1 f p B R D F q ( α i , α , β , λ ) d α d β
f p B R D F s = σ N S ( α α i ) 2 + 1 4 ( 1 α i 2 + 1 α 2 ) 2 ρ s , α 2     1 β 2 + 1 4 ( 1 + 1 β 2 ) 2 ρ p , β 2       × F s ( ( α + α i ) / 2 , n i k ) F p ( β / 2 , n i k )
f p B R D F p = σ N P ( α α i ) 2 + 1 4 ( 1 α i 2 + 1 α 2 ) 2 ρ p , α 2     1 β 2 + 1 4 ( 1 + 1 β 2 ) 2 ρ s , β 2       × F p ( ( α + α i ) / 2 , n i k ) F s ( β / 2 , n i k )
σ N 0 F q ( α i , n i k ) = 1 α 2 1 α 2 1 1 f p B R D F q ( α i , α , β , λ ) d α d β .
ρ q , α = ρ 0 ( 1 F q ( ( α + α i ) / 2 , n i k ) ) ( 1 F q ( 0 , n i k ) )
ρ q , β = ρ 0 ( 1 F q ( β / 2 , n i k ) ) ( 1 F q ( 0 , n i k ) )
f B R D F = f p B R D F s + f p B R D F p 2 .
S r = R ^ ( θ r ) f ^ p B R D F R ^ ( θ i ) S i
R ^ ( θ ) = ( 1 0 0 0 0 cos ( 2 θ ) sin ( 2 θ ) 0 0 sin ( 2 θ ) cos ( 2 θ ) 0 0 0 0 1 )
f ^ p B R D F = 1 2 ( f s + f p f s f p 0 0 f s f p f s + f p 0 0 0 0 2 f s f p 0 0 0 0 2 f s f p )
S s = ( g f s + β 2 f p g + β 2 ( g f s β 2 f p ) ( h α i 2 β 2 ) + 4 g h α i 2 f s f p β 2 ( g + β 2 ) ( h + α i 2 β 2 ) 2 ( h α i 2 β 2 ( g f s β 2 f p ) g β 2 f s f p ) ( h α i 2 β 2 ) ( g + β 2 ) ( h + α i 2 β 2 ) 0 )
S p = ( g f p + β 2 f s g + β 2 ( β 2 f s g f p ) ( h α i 2 β 2 ) 4 g h α i 2 f s f p β 2 ( g + β 2 ) ( h + α i 2 β 2 ) 2 ( h α i 2 β 2 ( β 2 f s g f p ) + g β 2 f s f p ) ( h α i 2 β 2 ) ( g + β 2 ) ( h + α i 2 β 2 ) 0 )
g = ( α 1 α i 2 + α i 1 α 2 β 2 ) 2
h = ( ( α 2 + β 2 ) 1 α i 2 + α α i 1 α 2 β 2 ) 2 .

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